Research

My major research interest can be classified into two broad domains viz. (1) Theoretical Statistics and (2) Statistical / Mathematical Ecology and epidemiology. My carrier gets initial thrust from researching the classical theoretical problems on statistics, specifically growth curve inference and goodness-of-fit test. Later on my interests shifted towards some of the fundamental and important aspects associated with ecology and epidemiology where mathematical modelling and statistical inferential procedure is extremely demanding with challenging datasets involving facet of experimental and field oriented studies.

Theoretical Statistics: Several attempts were made in developing various density dependent / independent growth curves applicable in various fields of research. But surprisingly the research based on “Growth Rates” and related statistical methodologies are ignored (Gupta et al., 2012) although it has immense utility in characterizing growth profiles of any species or physical quantity. Moreover, the lacunas in developing statistical inferential procedure when the growth responsible covariates are present in the model need to be explored through random effect structure. Apart from this in most of these studies involve measurement errors / reading errors which should be carefully modeled. Presently we are working on the exact distribution of relative growth rate (RGR), commonly known as a fitness function of species, frequently used in ecology and biological dynamics. Under both multivariate normal and log-normal structure of the size variable, we develop both parametric and nonparametric testing procedures for the equality of expected RGR profiles in two or multiple locations based on an ecological or biological size variable data on several species. Asymptotic and exact distributions of the proposed test statistics and its optimal properties are well illustrated.

Our present work also includes a study on the propagation of measurement error in biological experiments. If some assignable source of variation is present in the system, then any sample measurement will involve a significant amount of error and which can be transmitted with time. We are proposing metrics for the ‘propagation’ of error along with its estimates and the estimated variances. Asymptotics and other optimal properties are also studied.

“Goodness-of-fit” testing in growth curve model is another important arena where we are currently indulged. “Goodness-of-fit” test for Polynomial growth curve model was proposed by Hill (1968).We have extended such procedures for more generalized growth curve family such as exponential polynomial. In the spirit of the’proposed test, a similar procedure for testing “Goodness-of-fit” can be developed extended Logistic and Gompertz family.

Statistical / Mathematical Ecology: Relationship between per capita growth rate and abundance of a population has fundamental implications in population dynamics and different areas of ecology. Sibly et al. (Science, 2005) undertook an ambitious analysis of this problem by examining growth rates of 1780 time series of 674 species of four taxonomic groups, namely, birds, mammals, bony fishes and insects from Global Population Dynamics Database (NERC Centre for Population Biology, Imperial College, 2010).

Species growth models account for two apparently opposite factors that govern population dynamics: (1) the natural proclivity of the species population for exponential (Malthusian) growth, and (2) a negative density-dependence feedback governed by the environmental carrying capacity (K), which restrains population growth. However, the role of cooperation amongst conspecifics is a third factor that enhances population growth, and is generally ignored in currently available growth models. Cooperation in reproductive behaviours is likely to act as a positive densitydependent feedback to population growth (Avil´es, 1999). We consider cooperation as a fundamental aspect of population growth along with the other two factors explained above and propose an extended family of growth models. It is worthy to mention that the positive density dependent feedback mechanism from cooperative behaviour (e.g. cooperative feeding, cooperative breeding etc.) is reflected in the Allee effect (Allee, 1931; Odum and Allee, 1954) and our proposed model can be used to explain weak Allee mechanism. The proposed model can be further extended to explain strong Allee mechanism.

Our analysis reveals that a population with estimated cooperation parameter greater than an appropriate critical threshold may suffer from severe demographic threats. Bifurcation diagrams help us to identify populations where the estimated parameter is in the chaotic region and the population suffers from severe extinction risk. These findings have fundamental implications in extinction dynamics of species showing non-monotonic convex growth profile.